List of top Questions asked in GATE Electronics and Communication Engineering

Two resistors are connected in a circuit loop of area 5 m\(^2\), as shown in the figure below. The circuit loop is placed on the \( x-y \) plane. When a time-varying magnetic flux, with flux-density \( B(t) = 0.5t \) (in Tesla), is applied along the positive \( z \)-axis, the magnitude of current \( I \) (in Amperes, rounded off to two decimal places) in the loop is (answer in Amperes).

A 50 \(\Omega\) lossless transmission line is terminated with a load \( Z_L = (50 - j75) \, \Omega.\) { If the average incident power on the line is 10 mW, then the average power delivered to the load
(in mW, rounded off to one decimal place) is} _________.

In the circuit shown below, the AND gate has a propagation delay of 1 ns. The edge-triggered flip-flops have a set-up time of 2 ns, a hold-time of 0 ns, and a clock-to-Q delay of 2 ns. The maximum clock frequency (in MHz, rounded off to the nearest integer) such that there are no setup violations is (answer in MHz).

The diode in the circuit shown below is ideal. The input voltage (in Volts) is given by \[ V_I = 10 \sin(100\pi t), \quad {where time} \, t \, {is in seconds.} \] The time duration (in ms, rounded off to two decimal places) for which the diode is forward biased during one period of the input is (answer in ms).

X and Y are Bernoulli random variables taking values in \( \{0,1\} \). The joint probability mass function of the random variables is given by:
P(X = 0, Y = 0) = 0.06, \( \quad P(X = 0, Y = 1) = 0.14 \), \( \quad P(X = 1, Y = 0) = 0.24 \), \( \quad P(X = 1, Y = 1) = 0.56 \).
The mutual information \( I(X; Y) \) is (rounded off to two decimal places).

Consider the vectors \[ a = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad b = \begin{bmatrix} 0 \\ 3\sqrt{2} \end{bmatrix}. \] For real-valued scalar variable \( x \), the value of \[ \min_x \|a x - b\|_2 \text{ is} \quad \text{(rounded off to two decimal places)}. \] \( \| \cdot \|_2 \text{ denotes the Euclidean norm, i.e., for } y = \begin{bmatrix} y_1 \\ y_2 \end{bmatrix}, \quad \|y\|_2 = \sqrt{y_1^2 + y_2^2}. \)

Two fair dice (with faces labeled 1, 2, 3, 4, 5, and 6) are rolled. Let the random variable \( X \) denote the sum of the outcomes obtained. The expectation of \( X \) is _________ (rounded off to two decimal places).

Consider a system represented by the block diagram shown below. Which of the following signal flow graphs represent(s) this system? Choose the correct option(s).

All the diodes in the circuit given below are ideal. Which of the following plots is/are correct when \( V_I \) (in Volts) is swept from \( -M \) to \( M \)?

Consider a system where \( x_1(t), x_2(t), \) and \( x_3(t) \) are three internal state signals and \( u(t) \) is the input signal. The differential equations governing the system are given by: \[ \frac{d}{dt} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 0 \end{bmatrix} \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} u(t). \] Which of the following statements is/are TRUE?

Which of the following statements involving contour integrals (evaluated counter-clockwise) on the unit circle \( C \) in the complex plane is/are TRUE?

In the circuit shown, the identical transistors Q1 and Q2 are biased in the active region with \( \beta = 120 \). The Zener diode is in the breakdown region with \( V_Z = 5 \, V \) and \( I_Z = 25 \, mA \). If \( I_L = 12 \, mA \) and \( V_{EB1} = V_{EB2} = 0.7 \, V \), then the values of \( R_1 \) and \( R_2 \) (in \( k\Omega \), rounded off to one decimal place) are _________, respectively.

The intrinsic carrier concentration of a semiconductor is \( 2.5 \times 10^{16} \, /m^3 \) at 300 K. If the electron and hole mobilities are \( 0.15 \, {m}^2/{Vs} \) and \( 0.05 \, {m}^2/{Vs} \), respectively, then the intrinsic resistivity of the semiconductor (in \( k\Omega \cdot m \)) at 300 K is _________. (Charge of an electron \( e = 1.6 \times 10^{-19} \, C \))

A positive-edge-triggered sequential circuit is shown below. There are no timing violations in the circuit. Input \( P_0 \) is set to logic ‘0’ and \( P_1 \) is set to logic ‘1’ at all times. The timing diagram of the inputs \( SEL \) and \( S \) are also shown below. The sequence of output \( Y \) from time \( T_0 \) to \( T_3 \) is _________.

The identical MOSFETs \( M_1 \) and \( M_2 \) in the circuit given below are ideal and biased in the saturation region. \( M_1 \) and \( M_2 \) have a transconductance \( g_m \) of 5 mS. The input signals (in Volts) are: \[ V_1 = 2.5 + 0.01 \sin \omega t, \quad V_2 = 2.5 - 0.01 \sin \omega t. \] The output signal \( V_3 \) (in Volts) is _________.

Consider a real-valued random process \[ f(t) = \sum_{n=1}^{N} a_n p(t - nT), \] where \( T > 0 \) and \( N \) is a positive integer. Here, \( p(t) = 1 \) for \( t \in [0, 0.5T] \) and 0 otherwise. The coefficients \( a_n \) are pairwise independent, zero-mean unit-variance random variables. Read the following statements about the random process and choose the correct option.

(i) The mean of the process \( f(t) \) is independent of time \( t \).
(ii) The autocorrelation function \( E[f(t)f(t + \tau)] \) is independent of time \( t \) for all \( \tau \).

(Here, \( E[.] \) is the expectation operation.)

The \( Z \)-parameter matrix of a two-port network relates the port voltages and port currents as follows: \[ \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = Z \begin{bmatrix} I_1 \\ I_2 \end{bmatrix} \] The \( Z \)-parameter matrix (with each entry in Ohms) of the network shown below is _________.

In the circuit below, \( M_1 \) is an ideal AC voltmeter and \( M_2 \) is an ideal AC ammeter. The source voltage (in Volts) is \( v_s(t) = 100 \cos(200t) \). What should be the value of the variable capacitor \( C \) such that the RMS readings on \( M_1 \) and \( M_2 \) are 25 V and 5 A, respectively?

Consider a continuous-time finite-energy signal \( f(t) \) whose Fourier transform vanishes outside the frequency interval \( [-\omega_c, \omega_c] \), where \( \omega_c \) is in rad/sec.

The signal \( f(t) \) is uniformly sampled to obtain \( y(t) = f(t) p(t) \). Here, \[ p(t) = \sum_{n=-\infty}^{\infty} \delta(t - \tau - nT_s), \] with \( \delta(t) \) being the Dirac impulse, \( T_s > 0 \), and \( \tau > 0 \). The sampled signal \( y(t) \) is passed through an ideal lowpass filter \( h(t) = \omega_c T_s \frac{\sin(\omega_c t)}{\pi \omega_c t} \) with cutoff frequency \( \omega_c \) and passband gain \( T_s \).

The output of the filter is given by _________.

Consider the polynomial \[ p(s) = s^5 + 7s^4 + 3s^3 - 33s^2 + 2s - 40. \] Let \( (L, I, R) \) be defined as follows: \[ L \text{ is the number of roots of } p(s) \text{ with negative real parts.} \] \[ I \text{ is the number of roots of } p(s) \text{ that are purely imaginary.} \] \[ R \text{ is the number of roots of } p(s) \text{ with positive real parts.} \] Which one of the following options is correct?

Let \( G(s) = \frac{1}{10s^2} \) be the transfer function of a second-order system. A controller \( M(s) \) is connected to the system \( G(s) \) in the configuration shown below.

Consider the following statements.

(i) There exists no controller of the form \( M(s) = \frac{K_I}{s} \), where \( K_I \) is a positive real number, such that the closed-loop system is stable.
(ii) There exists at least one controller of the form \( M(s) = K_p + sK_D \), where \( K_p \) and \( K_D \) are positive real numbers, such that the closed-loop system is stable.

Which one of the following options is correct?

A 4-bit weighted-resistor DAC with inputs \( b_3, b_2, b_1, \) and \( b_0 \) (MSB to LSB) is designed using an ideal opamp, as shown below. The switches are closed when the corresponding input bits are logic ‘1’ and open otherwise. When the input \( b_3b_2b_1b_0 \) changes from 1110 to 1101, the magnitude of the change in the output voltage \( V_o \) (in mV, rounded off to the nearest integer) is _________.

All the components in the bandpass filter given below are ideal. The lower -3 dB frequency of the filter is 1 MHz. The upper -3 dB frequency (in MHz, rounded off to the nearest integer) is _________.

The generator matrix of a \( (6,3) \) binary linear block code is given by

The minimum Hamming distance \( d_{{min}} \) between codewords equals (answer in integer).

The function \( y(t) \) satisfies \[ t^2 y''(t) - 2t y'(t) + 2y(t) = 0, \] where \( y'(t) \) and \( y''(t) \) denote the first and second derivatives of \( y(t) \), respectively. Given \( y'(0) = 1 \) and \( y'(1) = -1 \), the maximum value of \( y(t) \) over \( [0, 1] \) is _________ (rounded off to two decimal places).